Parametric Study of Peripheral Nozzle Configurations

for Supersonic Retropropulsion

Noel M. Bakhtian∗

Stanford University, Stanford, CA, 94305

Michael J. Aftosmis†

NASA Ames Research Center, Moffett Field, CA, 94035

With sample-return and manned missions on the horizon for Mars exploration, the abil-ity to decelerate high-mass systems to the planet’s surface has become a research priority.This paper explores the use of supersonic retropropulsion, the application of jets facing intothe freestream, as a means of achieving drag augmentation. Numerical studies of retro-propulsion flows were conducted using a Cartesian Euler solver with adjoint-driven meshrefinement. After first validating this simulation tool with existing experimental data, aseries of three broad parametric studies comprising 181 total runs was conducted usingtri- and quad-nozzle capsule configurations. These studies chronicle the effects of nozzlelocation, orientation, and jet strength over Mach numbers from 2 to 8 and angles of attackranging from −5◦ to 10◦. Although many simulations in these studies actually producednegative drag augmentation, some simulations displayed local overpressures 60% higherthan that possible behind a normal shock and produced drag augmentation on the orderof 20%. Examination of these cases leads to the development of an aerodynamic model fordrag augmentation in which the retrojets are viewed as oblique shock generators. Flowapproaching the capsule face is decelerated and compressed by multiple oblique shocks.By avoiding the massive stagnation pressure losses associated with the bow shock in typi-cal entry systems, this approach achieves significant overpressure on the capsule face andstrong drag amplification. With a fundamental physical mechanism for drag augmentationidentified, follow-on studies are planned to exploit this feature and to understand its impacton potential entry trajectories and delivered mass limits for future Mars missions.

Nomenclature

A : area SUPERSCRIPTSCA : coefficient of axial force ∗ : nozzle throatCA,aeroshell : CA on the aeroshell (not incl. engine thrust)

′: non-dimensional

CA,forebody : CA on the capsule faceCA,total : coefficient of total axial force, Eqn. (8) SUBSCRIPTSCD : coefficient of drag b : capsule baseCL : coefficient of lift e : nozzle exitCP : coefficient of pressure o : stagnationCPoNS : stagnation CP behind normal shock at M∞ ∞ : freestreamCT : thrust coefficient, Eqn. (9)D : drag force GREEK SYMBOLSd : diameter α : angle of attackDaug : drag augmentation, Eqn. (10) β : ballistic coefficient, Eqn. (1)Dref : baseline (reference) drag value γ : ratio of specific heatsF : force θ : angular coordinate, Fig. 11J : exact functional value ρ : densityM : Mach number φ : nozzle tilt angle, Fig. 13

∗Ph.D. Candidate, Dept. of Aeronautics and Astronautics, Durand Building, 496 Lomita Mall, Student Member AIAA.†Aerospace Engineer, NASA Advanced Supercomputing Division, Senior Member AIAA.

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P : parametric space ABBREVIATIONSP : pressure AFL : Astrobiology Field LaboratoryPo1 : stagnation pressure in front of shock CFD : computational fluid dynamicsPo2 : stagnation pressure behind shock DGB : disk-gap-band (parachute)PoBS : stagnation pressure behind bow shock EDL : entry, descent, and landingPoNS : stagnation pressure behind normal shock JI : jet interactionPoOS : stagnation pressure behind oblique shock MSL : Mars Science Laboratoryq : dynamic pressure MSR : Mars Science Returnr : radius SRP : supersonic retropropulsionrnc : nozzle circle radius, Fig. 13T : thrust in axial directionu, v, w : velocity components

I. Introduction

Mars entry poses a challenge due to the planet’s thin atmosphere. With only about 1% of Earth’satmospheric density (Figure 1), Mars entry vehicles decelerate at much lower altitudes and have trouble

attaining sufficiently low terminal descent velocities without the aid of entry, descent, and landing (EDL)systems such as supersonically-deploying parachutes.1 To date, the United States has successfully landedsix systems on the surface of Mars, starting with the Viking I and II missions which landed in 1976. TheEDL system developed for these missions was based on a 70-degree sphere cone aeroshell and a supersonicdisk-gap-band (DGB) parachute. Amazingly, all subsequent missions including NASA’s 2011 Mars mission,the Mars Science Laboratory (MSL), have used only slight variants of this original EDL system.2

As mission requirements advance, target mass payload demands are increasing and are challenging thelimits of current EDL technology. Each of the six previous Mars missions landed at 0.6t or less, and theupcoming MSL mission has reached the landed payload mass capability (approximately 1t) of this Viking-based EDL system.2 Figure 2 shows the limits of the current DGB parachute deployment region and theeffect of increasing β, the ballistic coefficient. This value, characterizing the response of an entry vehicle toatmospheric braking, scales directly with mass and is given by

β =m

CDA(1)

where m is the system’s mass, CD is the hypersonic drag coefficient, and A is the reference area. As seenin Figure 2, with increasing payload mass (and thus β), the max-Mach deployment limit of 2.1 for the DGBsystem is reached at increasingly lower altitudes. For reference, only Mars entry systems with β less thanapproximately 50kg/m2 are able to reach the subsonic propulsion region without supersonic decelerationmethods, and all previous missions have ranged from β = 62− 90kg/m2.1

Proposed future missions such as the robotic Mars Sample Return (MSR) and the Astrobiology FieldLaboratory (AFL) would result in even higher touchdown masses, and the requirements of human missions toMars could demand 40-100t payloads, a 2-order of magnitude increase over previous landed mass capability.2

Figure 1. Atmospheric density comparisonfor Earth and Mars.

With these increasingly aggressive payload demands on thehorizon, it is evident that the approximate ballistic coefficientlimit of the heritage DGB parachute technology, 150kg/m2,will be quickly exceeded. The challenges imposed by the largemass requirements of future Mars exploration necessitate thedevelopment of additional EDL technologies. Proposed solu-tions for adequate deceleration include a) a reduction of theballistic coefficient through an increase in reference area, b)the development of supersonic parachutes capable of deploy-ing at higher Mach number, c) an increase in vertical lift, ord) the development of a new decelerator such as supersonicretropropulsion.1

This paper examines the development of supersonic retro-propulsion (SRP) as a means of deceleration. Historically,

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Mach 1 Mach 2 Mach 3

velocity (m/s)

altitud

e (km

MO

LA

)

25

20

15

10

5

700600500400300200100

!=25

!=50

!=100

!=200

!=150

Subsonic

Propulsion

Region

Subsonic Parachute

Deployment or

Heat Shield

Separation Region

Supersonic

Parachute

Deployment

Region

Figure 2. EDL method deployment regions, showing β [kg/m2] limits. Adapted from Ref. 1.

retropropulsion has involved the use of thrusters directed into a subsonic freestream flow, the aim of whichis to produce a thrust force to supplement the aerodynamic drag, thus providing increased decelerationby increasing the total force on the body. Given the increasing velocities of Mars entry, extension of thisdeceleration technique to supersonic freestream has been proposed. Thus far, preliminary SRP developmenthas focused on increasing engine thrust while attempting to maintain the bow shock about the entry bodyfor supplemental drag.3 The deceleration provided using this strategy is highly dependent on the mass offuel transported solely for atmospheric entry. In this study, we consider an alternative role, where super-sonic retropropulsion is used as a mechanism to increase aerodynamic drag by achieving significant surfaceoverpressures rather than merely rely on fuel transport for terminal thrust augmentation.

The bow shock, considered a key decelerative physical mechanism due to the post-shock pressure increase,also drastically reduces the post-shock stagnation pressure on the body. This means that the maximumrecoverable pressure on the capsule face is severely reduced. For example, at M∞ = 6, the stagnationpressure behind a normal shock has decreased to less than 3% of its initial value (PoNS/Po1 = 0.02965),meaning that the maximum pressure one can recover behind such a shock has dropped by 97%. Recognizingthis, the current study seeks to exploit the available reservoir of potential drag by altering the flow physicsof the system through the use of retropropulsive jets.

Very little data has been collected on retropropulsive jets in a supersonic freestream.3 The majority ofresearch in this area of “counterflowing”, “opposing”, and “forward-facing” jets took place as experimentalstudies in the 1960’s and concentrated on a central single-nozzle configuration.4–8 The aim of these studieswas to analyze the effectiveness of a retrojet as a means of reducing drag and surface heating. With thesame emphasis on drag reduction, numerous numerical studies on central nozzle configurations have beenperformed as well.9–13 These studies show that a single opposing central jet acts as an aerospike, significantlyreducing the aerodynamic drag which is the opposite effect of that required by SRP for deceleration.14

In contrast to these studies on drag-reducing central single-nozzle configurations, there is a lack of researchon alternate configurations that might prove useful to the SRP community. The most prominent in theliterature are experimental multiple-nozzle configuration observations made by Jarvinen and Adams,15,16

Keyes and Hefner,17 and Peterson and McKenzie,18 which suggest that certain multiple-nozzle configurationsmaintain the drag resulting from the bow shock in addition to providing deceleration from retrothrust, thusgiving some additional retroforce. These studies, although increasing the entire axial force with supplementalthrust, show no increase in aerodynamic drag when compared to the baseline (no SRP) case, leading to atotal reliance on the mass of fuel transported for this purpose.

This paper provides a series of broad numerical parametric studies to build a preliminary understanding ofthe physical mechanisms involved in multi-nozzle (peripheral) SRP flows. Section II describes the flow solverand grid refinement approach used in this study. Section III offers a brief comparison with experimentaldata and provides an analysis of the tri- and quad-nozzle parametric studies, leading to the development ofa flow model for SRP drag augmentation.

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II. Computational Approach

This study was accomplished using the steady-state simulations and adaptive mesh refinement of theCart3D simulation package. This section briefly describes the Cartesian-based Euler solver and providesan overview of the adjoint-based mesh refinement module. Finally, applicability of the steady-simulationapproach to SRP flows is examined through a comparison with a relevant high-resolution, time-accuratesimulation.

A. Numerical Method

Simulations for this investigation are performed using the Cart3D package. This method uses a parallel,multi-level Euler solver on automatically-generated Cartesian meshes with embedded, cut-cell boundaries.Cart3D, developed by Aftosmis and Berger in References 19 and 20, has recently been extended to includean adjoint-based mesh adaptation method to guide cell refinement and control discretization errors presentwithin complex flow fields.21–23

Following the creation of a coarse starting mesh upon which the initial flow solve is computed, a cell-wise error-estimate is calculated using the method of adjoint-weighted residuals.21,24 The mesh is refinedaccording to a “worst things first” strategy, and the process repeats until a minimum error or maximum cellcount (user-defined) is reached.

The spatial discretization of the Euler equations uses a cell-centered, second-order accurate, upwindfinite-volume method with a weak imposition of solid-wall boundary conditions. Steady-state flow solutionsare obtained using a five-stage Runge-Kutta scheme with local time stepping, using the flux-vector splittingapproach of van Leer and multigrid convergence acceleration.19,20,25–27 The solution algorithm for theadjoint equations utilizes the same parallel, multi-level framework as the base Euler solver.21,22

Widely used for producing aerodynamic databases in support of engineering analysis and design due to itsrobustness and speed, this inviscid simulation package suits the parametric nature of this study.28–30 More-over, this work focuses on simulations with strong and unfamiliar off-body flow features, making it difficultto manually create appropriate meshes a priori. The adjoint-based mesh adaptive approach described hereallows automation of the meshing process, and includes information on mesh convergence and discretizationerror with each simulation.

B. Adjoint-based Mesh Refinement

As simulation geometries and flows become more complex, the ability to create a suitable mesh a prioripresents an increasing challenge. Parametric and optimization studies amplify this problem by requiringhundreds of potentially unique meshes. We address this need through use of an output-based mesh adaptationmethod. The following discussion outlines the key steps of this method. Further details are provided inReferences 21–23.

Number of Cells10

410

510

610

710

810

9

Funct

ional

(C

L, C

D, ...)

J(UH)

E

Exact Solution

Approximate

Functional

Number of Cells10

410

510

610

710

810

9

Funct

ional

(C

L, C

D, ...)

J(Uh)

e

J(UH)

J

J(QH)

J(Qh)E

e

Figure 3. Convergence of functionalJ with mesh refinement, showing thedefinition of total error, E, and relativeerror, e.

Here, the end goal of the simulation is to provide a reliable ap-proximation of a functional J (Q), such as lift or drag, that is itselfa function of a flow solution Q = [ρ, ρu, ρv, ρw, ρE]T satisfying thesteady-state, three-dimensional, discretized Euler equations of a per-fect gas,

R(QH) = 0 (2)

Using J() to represent the discrete operator used to evaluate thefunctional, J(QH) now denotes the approximation of the functionalcomputed on an affordable (coarse) Cartesian mesh with characteris-tic cell size H, where Q = [Q1, Q2, · · · , QN ]T is the discrete solutionvector of the cell-averaged values for all N cells of the mesh.

Figure 3 shows convergence of a model aerodynamic functionalwith mesh refinement, where E is the total error in this output dueto discretization error in the numerical solution. This error is definedas the difference between the exact functional value and the valueobtained when evaluating this functional using the discrete solutionon the current mesh, E = |J (Q)−J(QH)|. Rather than attempt tocompute E directly, we use the approach of Venditti and Darmofal

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(Ref. 24) and consider the simpler problem of estimating how the discrete functional evaluation J(QH) wouldchange if solved on a finer mesh, h. The relative error, sketched in Figure 3, is defined as the differencebetween the functional evaluations on the current mesh H and the finer, embedded mesh h,

e = |J(Qh)− J(QH)| (3)

For a second-order method on a sufficiently smooth solution in the asymptotic range, knowing the relativeerror gives total error in the output as

E = e+14e+

142e+ · · · = 4

3e (4)

Knowledge of the relative error e, however, hinges on the ability to evaluate the functional using the finemesh solution, J(Qh). Without solving on this finer mesh, an estimate of the functional on the embeddedmesh can be found using Taylor series expansions of the functional and residual equations about the coarsemesh solution, given by

J(Qh) ≈ J(QH

h

)−(ψH

h

)TR(QH

h

)︸ ︷︷ ︸AdjointCorrection

−(ψh − ψH

h

)TR(QH

h

)︸ ︷︷ ︸RemainingError

(5)

where R() is the spatial operator of the Euler solver, ψ is the discrete adjoint soution, and the notation()H

h indicates prolongation from the coarse to fine mesh.24 The adjoint variables satisfy the following linearsystem of equations: [

∂R(QH)∂QH

]T

ψH =∂J(QH)∂QH

T

(6)

Details on the adjoint solver can be found in Ref. 22, and Refs. 21 and 31 contain verification and validationstudies on this error-estimation method.

In Equation (5), the adjoint variables provide two correction terms, one which improves the accuracyof the functional on the coarse mesh, and one serving as a remaining error term which is used to form anestimate of each cell’s contribution to the error in the objective function. Convergence of the calculatedfunctional towards its true value occurs as this remaining error term decreases, making the cell-wise error-estimate the focus of the adaptation strategy. This term depends on the solution of the adjoint equationon the embedded mesh, ψh, but this can be approximated with an interpolated adjoint solution from thecoarse mesh.32 Note that the remaining error is formed through an inner product involving both the flowresidual of the prolonged discrete solution (essentially an estimate of local truncation error) and the adjointinterpolation errors (serving as a weighting term). Both of these terms need to be large for cell refinementto improve the functional output, J(Q).

Figure 4. A pitch-plane central cut showing mesh refinement resulting from the automatic adjoint-basedmethod. The initial mesh (left) contains 23,000 cells, and the mesh after five adapt cycles (right) contains763,000 cells.

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Figure 5. Convergence of the functional and error-estimate.

In a typical case using adjoint-based mesh refinement, the corrected functional will lead the computedfunctional J(QH) towards the true functional value J (Q) as the number of cells increases. Each of thesecycles should see the remaining error value drop as the mesh is refined and J(QH) converges. Figure 4 showsa generic initial mesh (23,000 cells) and an automatically-adapted final mesh (763,000 cells) for one of thecasesa in the parametric study of Section IIIC. The functional appears to be approaching an asymptotic valueby the fifth adaptation cycle despite the complex flow features, as seen in Figure 5. This is corroboratedby the drop in the functional’s error-estimate (the cell-wise error summed over all cells), seen in the samefigure.

As discussed in Refs. 32 and 31, the error-estimate also reveals the refinement level at which this steadymodeling approach begins to break down. An increase in the error-estimate in the final adaptation cycleindicates “unsteadiness,” or noise due to incomplete flow convergence. The next section compares resultsobtained with the described steady method to those from a time-accurate simulation of relevant SRP physics,supporting the use of steady simulations in this preliminary study.

C. Comparison with Time-Accurate Simulations

Before proceeding with extensive parametric studies, it is prudent to examine the utility of the describedsteady and inviscid model when applied to flow physics represented in SRP cases. Examples of comparisonsbetween steady Cart3D simulations and bluff-body time-accurate simulations are given in Reference 32. Wealso give a brief comparison with SRP time-accurate simulations here, and provide a comparison againstrelevant experimental data in Section IIIA.

Full unsteady simulations were run using a time-accurate version of the Cartesian Euler solver,

Figure 6. A pitch-plane central cut throughthe mesh (4.5 million cells) used for time-accurate simulation comparisons.

developed in Ref. 33. This solver uses the same multi-level,parallel framework with time accuracy achieved through a dual-time formulation in which multigrid is used to converge innerpseudo-time iterations. Figure 6 contains the 4.5 million cellmesh used for both steady and time-accurate simulations. Thismesh was generated by further adaptation of the mesh shownin the previous section.

Figure 7 gives a quantitative comparison between the twosimulations by showing the convergence histories of forces onthe capsule for the steady simulation as a function of multigridcycles in the frame on the left and for the unsteady simulationon the right. The averaged drag coefficient values for the steadyand time-accurate cases are 1.520 and 1.530, respectively. Theunsteady simulation required a non-dimensional timestep of0.006b, and was initialized with 200 cycles of multigrid relax-ation to produce the time-accurate results in Figure 7. This isroughly a factor of 50 times more computational effort than thesteady results shown adjacent. Given both the close agreement

aCase: quad-nozzle, M∞ = 6, α = 0◦, rnc = 0.85, φ = 0◦, CT = 0.5bThe non-dimensional time is based on the freestream speed of sound and the capsule diameter.

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Steady-state Simulation Time-dependent Simulation

Figure 7. Comparison of force coefficients using steady-state and time-dependent approaches. Computedaverages and averaging windows are shown with dotted lines, and force scales are identical in both frames tofacilitate comparison.

between the time-averaged coefficients and the steady-state results and this wide disparity in computationalcost, we used adaptively-refined steady-state modeling for the simulations in the parametric studies con-ducted in the present work.

III. Numerical Simulations and Discussion

The presentation of numerical simulation results opens with a validation study supporting use of thesteady, inviscid method described in Section II to study problems involving counterflowing jets in a supersonicfreestream. A series of parametric studies then highlights the effects of retrojet nozzle placement and jetstrength on drag coefficient. These studies increasingly focus on examples which exhibit a substantial capsuleface overpressure and establish trends valuable for future design studies. Finally, we introduce a SRP flowmodel describing a physical mechanism leading to significant drag augmentation, supported by data fromthe parametric studies.

A. Comparison with Experimental Data

This section examines the use of the described numerical method for approximating the complexities of SRPflow through a comparison with experimental SRP data containing the relevant physics. An experimentaldataset given in a study by Jarvinen and Adams (Ref. 15), representing one of the few published surveys onperipheral SRP configurations, is used here to compare force and pressure data.

1. Geometry and Simulation Details

The current simulations use geometries taken from the experimental studies with tri-nozzle and single-nozzle aeroshell configurations, given in Ref. 15. The experiments simulated in this study were conductedon a 60◦ half angle, conical aeroshell, 101.6mm in diameter (4in). Figure 8 and Table 1 give furthergeometric details of the tri- and single-nozzle configurations. The three 15◦-conical, equally-spaced nozzlesof the peripheral tri-nozzle geometry (shown in Figure 8) are scarfed at 30◦ by the aeroshell surface. Thesenozzles are placed such that the center of the throat lies at 0.80rb from the aeroshell’s line of symmetry.The 15◦-conical nozzle of the central single-nozzle geometry is placed on the aeroshell’s line of symmetry.

Table 1. Geometry specifications for thetwo configurations used in the experi-mental comparison study.

tri-nozzle single-nozzledb(mm) 101.6 101.6Ab/Ae 192.77651 64.25884Ae/A

∗ 13.95 13.95

The final simulation geometries were discretized into 118,000 and123,000 triangles for the single- and tri-nozzle configurations, re-spectively.

For this validation study, the baseline tri- and single-nozzleconfigurations were run at M∞ and α conditions tested experi-mentally in Ref. 15. Simulation power boundary conditions, P ′,ρ′, and u′, v′, and w′ were applied at each nozzle throat. We usedthe incremental mesh refinement method of Section IIB, target-ing error in the functional chosen as the simple weighted sum of

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lift and drag coefficients on both the capsule face and the engines,

J = 0.8(CL + CD)|engines + 0.2(CL + CD)|capsuleface (7)

Approximately 5 adapt cycles were used in each case, resulting in mesh sizes on the order of 400,000 cells.

2. Comparison of Force and Pressure Data

30°

15°

d* = 1.9592mm

60°rb = 50.8mm

rnc = 0.8rb

tri-nozzle

capsule face

aeroshell

engine

de =

7.3

17

5m

m

d* =

3.3

93

4m

m

de =

12

.67

44

mm

rn = 0.2rb

r s =

0.0

6r b

Figure 8. Geometric details of the tri-nozzle and single-nozzle model geometries. Colored outlines illustrate the por-tions of the surface included in the “capsule face” (cyan),“aeroshell” (blue), and “engine” (red) force integrations.

The experimental work in Reference 15 focusedon variations of the total axial force coefficientCA,total, which is a sum of the axial thrust co-efficient and the axial surface forces acting onthe aeroshellc

CA,total = CT + CA,aeroshell (8)

Figure 9 shows the numerical and experimen-tal variation in CA,total and CA,forebody withthrust coefficientd,

CT =T

q∞Ab(9)

for both the single-nozzle and tri-nozzle ge-ometries at M∞ = 2 and α = 0◦. Ex-perimental values were lifted from Figures 55and 56 of Reference 15. Tri-nozzle resultsare represented by triangles while circles rep-resent single-nozzle data points. The blackline in Figure 9a indicates the values at whichCA,total = CT , meaning that the entire axialforce on the model is due to the thrust aloneand that aerodynamic drag is negligible. Ex-amining the single-nozzle cases in Figure 9a,total axial force is seen to immediately drop tothe thrust level by CT = 0.5. The tri-nozzlecases maintain a higher total axial force at lowCT , but this too drops down, asymptoting tothe levels established by the thrust by CT = 3.Numerical data follows these trends closely.

The discrepancy in tri-nozzle data seen athigher CT levels is better examined in Figure 9b, which highlights the drag value itself (the force level abovethe black line in Figure 9a), by plotting CA,forebody as a function of CT . Here, we can see that the numericaltri-nozzle values do drop to match experimental data at CT ≥ 4. This lag in drag reduction with CT ismirrored, to a lesser extent, in the single-nozzle data as well. Numerical results at CT = 0e are seen to agreewith experimental data to within 4%. Of note is the complete lack of aerodynamic drag on the capsule facefor the single-nozzle configuration at CT > 1 and the tri-nozzle configuration at CT > 3. This suggests that,at these conditions, a single jet or set of jets (each at CT > 1) actually acts as a drag-resistant aerospike bypreventing high pressure from developing on the capsule face.

A similar comparison was also completed both over a range of α values and for M∞ = 1.05 and 1.5. Ingeneral, the same trends are seen in both numerical and experimental datasets, and numerical values give a

cDue to experimental restrictions, accurate pressure measurements could not be taken over the entire capsule base. As aresult, the experimental CA,aeroshell was evaluated assuming that freestream static pressure was acting over the entire modelbase, giving forebody axial force coefficient, CA,forebody . In order to compare with the experimental data, the numerical datawas post-processed with the same technique. The numerical data in Figure 9 was computed using a) CA,total = CA,forebody+CT

and b) CA,forebody .dDelivered thrust coefficient is typically different than requested thrust coefficient due to nozzle scarfing and cosine losses.eFor this study, jets-off cases were run on geometries with nozzles to be consistent with Reference 15. In contrast, the

parametric studies of Sections IIIB and IIIC utilize a clean aeroshell (no nozzles) for baseline cases.

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a) Total axial force coefficient b) Forebody axial force coefficient

Figure 9. Comparison of experimental data from Ref. 15 (magenta) and numerical results (blue) for bothsingle- and tri-nozzle configurations at M∞ = 2 and α = 0◦.

high confidence in the simulation model. The tri-nozzle case at M∞ = 2 and α = 0◦ with CT = 1 is furtherhighlighted in Reference 15 because it led to one of the highest drag values in the study. Figure 10 showsnumerical surface pressure data compared with data taken from experimental pressure taps on the modelface (lifted from Figure 72 of Reference 15). Experimental pressure coefficients are given in the circles shownat the locations of the respective pressure taps, while the numerical pressure coefficient distribution is shownover the remainder of the capsule face. Surface pressure magnitudes show very good qualitative agreementover the capsule face.

Figure 11 gives a more quantitative view, comparing surface pressure profiles along radial directions (fromnose to periphery) for the same case, with experimental data lifted from Figure 66 in Reference 15. The blackline along Cp = 0 represents values at which measured pressure is equal to freestream static pressure. Thedata are taken along radial lines (constant θ), the positions of which are portrayed in the inset of Figure 11.This figure shows the numerical results closely following the experimental trend along each of the radial lines.Radial lines passing between the nozzles (θ = 90◦ and θ = 180◦) show a maximum pressure at the center ofthe capsule face with a gradual decrease in the direction of the capsule periphery. The θ = 90◦ line, lying 30◦

closer to a nozzle, maintains a higher pressure across more of the capsule face than the θ = 180◦ line, but

Cp 2.5

2.0

1.5

1.0

0.5

0

-0.5

Figure 10. Surface pressure comparison forthe M∞ = 2, α = 0◦, CT = 1 case. Exper-imental CP data (Ref. 15) is given withinthe circles (at the locations of the respectivepressure taps) and the numerical CP distri-bution is shown over the remainder of thecapsule face.

!

Figure 11. Comparison of radial pressure profiles over thecapsule face for the M∞ = 2, α = 0◦, CT = 1 case. Lines fordata extraction at θ = {0◦, 90◦, 180◦} are shown on the sketchof the capsule face inset above the legend.

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Figure 12. Numerical CD and Daug data for M∞ = 2 and α = 0◦, showing tri- and single-nozzle comparison.For reference, the dashed line indicates Daug = 0.

experiences lower pressures in the vicinity of the nozzle. The θ = 0◦ line actually cuts through a nozzle, andso hits a maximum pressure on the surface prior to the nozzle. The pressure coefficient behind the nozzle(between the nozzle and the capsule edge) is negative, due to the wake behind the plume. Consistent withFigure 10, numerical data follows the experimental trends closely, but experiences slightly higher pressuresalong radial lines between nozzles. The similarities in both values and trends between numerical simulationdata and the experimental studies of Reference 15 strongly reinforce our confidence in the simulation abilityto capture the relevant SRP flow physics using this inviscid, steady modeling approach.

This analysis, revealing that the beneficial drag force on the body disappears at tri-nozzle CT levelsgreater than 3, implies that use of SRP in the manner of Ref. 15 would be severely limited as a decelerativetechnique. First, any system requiring more thrust than CT = 3 would need to rely entirely on the thrust fordeceleration, since aerodynamic drag is negligible in that range. Second, any system relying on the addedbenefit of drag in addition to thrust would be limited to operating at CT levels less than 3.

Moreover, it is clear from Figure 9b that SRP (CT > 0) in the experimental study consistently decreasesthe drag experienced by the bodyf. This can be expressed as a negative drag augmentation, with dragaugmentation due to jet interaction defined as

Daug =D −Dref

Dref(10)

where the reference drag value here is the drag experienced by a capsule with no SRP (CT = 0) at the sameM∞ and α. With this definition, doubling the baseline drag gives Daug = 1, maintaining the baseline draggives Daug = 0, and elimination of all drag results in Daug = −1. We see this clearly in Figure 12, whichshows the numerical results of this study plotted as CD and the corresponding drag augmentation, Daug,with the CT = 0 values for the tri- and single-nozzle configurations used as Dref in Eqn. (10). Again it isnoted that all of the cases based on the experimental study experienced negative drag augmentation, or alower drag than achieved by a capsule without SRP. While it is obvious that the deceleration still increaseswith the thrusting force regardless of the lack of aerodynamic drag, it seems ironic to rely on fuel storageand transport to solve an EDL mass problem. Rather than depend on thrust for deceleration, this studyaims to introduce a SRP mechanism by which drag itself is augmented, using a limited amount of thrust.In the absence of further significant experimental SRP data, and to explore any possible mechanisms whichmay lead to significant drag augmentation, several parametric studies were executed and are described inthe next sections, varying jet strength, placement, and orientation over a range of wind-space conditions.

B. Initial Parametric Study

With the lack of experimental and simulation data from peripheral nozzle configurations over a range of condi-tions and geometries, a parametric study is performed here to establish a baseline understanding of the physi-cal intricacies of supersonic retropropulsion and to determine the potential for drag augmentation. This studyuses the Jarvinen and Adams tri-nozzle geometry as a starting point. First, a baseline study is completed,

fThe references to “force amplification” in the peripheral tri-nozzle cases in Reference 15 allude to increases in total axialforce due to thrust addition.

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rnc

M!

Figure 13. Variables testedin the parametric study:nozzle location (rnc) as afraction of rb, nozzle tilt (φ),capsule angle of attack (α),and freestream Mach num-ber (M∞).

varying nozzle placement, orientation, and jet strength at M∞ = 2 and α = 0◦.An augmented study follows, extending the analysis to higher freestream Machnumbers.

1. Geometry and Parameterization

The baseline tri-nozzle geometry used for these parametric studies is basedon the configuration used by Jarvinen and Adams (Ref. 15). The aeroshellgeometry is the same as that created for the validation study (described inSection IIIA.1). The three 15◦ conical nozzles, equally-spaced and scarfed bythe aeroshell surface, have an Ae/A∗ = 4.23 with Ab/Ae = 200.

The parameter space includes variations in nozzle placement and orienta-tion, jet strength, and wind-space conditions. Figure 13 shows the geometricnozzle parameters (the nozzle circle radius, rnc, and the nozzle tilt, φ) andFigure 14 shows each of the 12 nozzle configurations. These geometric param-eters are varied over a range of freestream Mach numbers, capsule angles ofattack, and jet thrust coefficients (defined in Eqn. (9)). In total, we examinea 5-dimensional parametric space P = P(M∞, α, rnc, φ, CT ). Evaluated para-metric values are listed in Table 2. In all, 103 simulation cases were run. Thebaseline tri-nozzle study tests the jet parameters at M∞ = 2 and α = 0◦. Addi-tional wind-space conditions are then tested in the augmented tri-nozzle studyusing a subset of the configurations.

nozz

le c

ircl

e ra

diu

s, r

nc

nozzle tilt angle,

Figure 14. Geometric configurations tested in the initial (tri-nozzle) parametric studies. Shown are variationsin nozzle circle radius, rnc = {0.20, 0.50, 0.85}, and nozzle tilt angle, φ = {0◦, 30◦, 45◦, 60◦}.

2. Baseline Tri-nozzle Study

Table 2. Range of testedgeometric, boundary con-dition, and wind-spaceparametric values for thetri-nozzle studies.

M∞ 2, 4, 6, 8α 0◦, 10◦

rnc 0.2, 0.5, 0.85φ 0◦, 30◦, 45◦, 60◦

CT 0, 0.5, 1, 3, 6

The baseline parametric study consisted of cases varying nozzle placement,orientation, and jet strength over the entire range given in Table 2 atM∞ = 2g and α = 0◦. In total, 49 cases were run for this study, includingthe reference jets-off case. This CT = 0 case resulted in a drag coefficientof 1.52 against which all of the retropropulsive cases in this section arecompared. The drag augmentation results (Eqn. (10) with the CT = 0case taken as the reference) are given in Figure 15. As described in SectionIIIA.2, drag augmentation gives the relative change in drag as comparedto a configuration with no SRP.

Each plot in Figure 15 compares drag augmentation for capsules withvarious nozzle positions (rnc) as a function of thrust coefficient, with noz-zle tilt (φ) varying between each plot. The rnc = 0.2 cases, those withnozzles located closest to the center of the capsule, tend to fare the worst,

gChosen because the best results were achieved in the Jarvinen and Adams experimental work at this Mach number.

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Dra

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ug

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au

g

Dra

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ug

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tio

n, D

au

g

Thrust Coefficient, CT

Dra

g A

ug

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n, D

au

g

Dra

g A

ug

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n, D

au

g

= 0° = 30°

= 45° = 60°

Thrust Coefficient, CT

Thrust Coefficient, CT Thrust Coefficient, CT

Figure 15. Drag augmentation values for the baseline tri-nozzle study (M∞ = 2, α = 0◦).

with drag falling towards 0 with increasing thrust coefficient. Negative drag, or control reversal due to jetinteraction, is even experienced at φ = 30◦ and CT = 6. In this case, the centrally-placed jets actually act asan aerospike and reduce the drag of the baseline system. The rnc = 0.85 cases, however, with nozzles locatedat the periphery of the capsule, maintain the highest drag even at the larger thrust coefficients (seeminglyindependent of CT and φ) by holding the plume away from the capsule pressure face, especially at non-zerotilt angles, thus allowing unhindered flow from the bow shock to affect the surface pressure.

The low surface pressures experienced by cases with centrally-located nozzles are the result of wake flowor the plume flow itself. Figure 16 aids in visualizing this low-pressure effect by portraying the differences dueto plume position for a set of extreme cases, P(M∞, α, rnc, φ, CT ) = P(2, 0◦, {0.20, 0.50, 0.85}, 60◦, 6). Thetop row of images in Figure 16 show pressure coefficient on the body and Mach contours on a cutting-planepassing through the center of the capsule and intersecting a nozzle. The color map has been chosen to aid inflow comprehension, with white representing Mach 1 contours, blue representing subsonic regions, and yellowto red representing supersonic areas. The lower row of images show the corresponding capsule face pressurecoefficients. Here, the color map facilitates visualization of overpressure, with white representing CPoNS

(the stagnation pressure coefficient behind a normal shock at M∞), blue representing underpressure, andyellow to red showing areas of overpressure. From this figure, it is clear that nozzles located at the capsuleperiphery allow higher pressures over the majority of the capsule face, although no areas of overpressure onthe capsule face are evident or expected here.

The only cases in which the peripheral jets decreased the drag significantly were those performed at anozzle tilt of 0◦ and high CT . When the heavy plume flow runs head-on into the freestream flow, a huge wakeregion forms behind the plume, directly affecting the capsule face and obviating any high surface pressures.At higher tilt angles, the capsule face no longer lays in the wake of this interaction region, allowing thesurface to maintain its post-bow shock surface pressures and thus maintain its baseline drag value.

The major observation in this study is the lack of drag augmentation in a majority of the cases. Atbest, the peripheral jets manage to maintain the drag obtainable without retropropulsion (just a capsulewith bow shock), while a majority of the more centrally-located jet configurations actually resulted in asevere decrease in drag as compared to this reference. The three cases which marginally augmented thedrag, giving maximum Daug values of 0.030 (a 3% drag increase), did so by simply straightening the bowshock (Figure 17) such that the freestream flow experienced a lengthened section of normal shock in front

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3.6

2.7

1.8

0.9

0

CP

2.0

1.5

1.0

0.5

0

CP

5

4

3

2

1

0

Mach

Figure 16. Flow images showing effect of nozzle location: P(M∞, α, rnc, φ, CT ) = P(2, 0◦, {0.20, 0.50, 0.85}, 60◦, 6),from left to right. Plume and shock visualization (upper frame) is shown through a pitch-plane central cut(flow from left to right), with the Mach contour color map chosen such that the sonic line is white, supersonicflows are yellow and red, and subsonic flows are blue. CP is shown on the capsule. Also pictured are capsuleface CP distributions (lower frame) with the color map chosen such that white indicates CPoNS, yellow andred indicate overpressure, and blue indicates underpressure.

of the capsule face, leading to slightly increased pressures on the face. With the plumes caught between thebow shock and the body, this is the only mechanism by which the capsule face can experience an increasein pressure (and thus drag) because the maximum surface pressure is determined by the strength of theshock. However, the stagnation pressure loss due to the bow shock severely limits the possibility of dragaugmentation. As described in Section I, the stagnation pressure loss behind a normal shock at M∞ = 2 isalmost 30% and drastically increases at higher M∞. Previous studies have chosen to focus on these casesdespite this, using the engine thrust as a possible means of further deceleration. This brute force method,however, is based on and limited by the available fuel. The remainder of this study seeks to achieve an alter-nate means of increasing the deceleration by tapping into this stagnation pressure potential through shock

2.0

1.5

1.0

0.5

0

CP

5

4

3

2

1

0

Mach

Figure 17. Visualization of the lengthening of thenormal shock portion of the bow shock, causingminimal drag augmentation: P(M∞, α, rnc, φ, CT ) =P(2, 0◦, 0.85, 60◦, {0, 1, 3}). The CT = 3 case (right) ex-perienced the highest drag augmentation values in thisstudy (Daug = 0.030).

structure manipulation, thus allowing surface over-pressures and increased drag. An augmented studyis completed next, enlarging the parameter space toinclude higher Mach numbers and non-zero angles ofattack to establish the potential for significant levelsof drag augmentation.

3. Augmented Tri-nozzle Study

To investigate trends at higher freestream Machnumbers, the study of Section IIIB.2 is ex-tended to include M∞ = 4, 6, 8. This studyexamines cases located along the edges of thehypercube of the parametric space given inTable 2, P(M∞, α, rnc, φ, CT ) = P({4, 6, 8},{0◦, 10◦}, {0.20, 0.85}, {0◦, 60◦}, {1, 6}). In total,54 cases were run in this study including sixjets-off (CT = 0) cases used for reference.Table 3 gives CD for the six jets-off cases.

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Table 3. CT = 0 (jets-off) reference values:CD over a range of M∞ and α.

M∞ = 4 M∞ = 6 M∞ = 8α = 0◦ 1.473 1.465 1.465α = 10◦ 1.411 1.390 1.379

5

4

3

2

1

0

Mach

3.6

2.7

1.8

0.9

0

CP

Figure 18. Mach contours (left) and capsule face CP (right)for P(M∞, α, rnc, φ, CT ) = P(4, 0◦, 0.85, 0◦, 1).

Drag augmentation values for each casein the study are given in Figure 19. TheDref used to determine Daug for each run(Eqn. (10)) is given by the specific CT = 0drag value for the corresponding combinationof Mach-α. Similar to the previous study, allcentral nozzle configurations (rnc = 0.20) decrease the drag by blocking the capsule face with the plumes,even at higher nozzle tilt angles. However, in this augmented study, several of the peripheral nozzle cases(rnc = 0.85) do exhibit substantial drag augmentation (up to 0.218, a 21.8% drag increase).

Figure 18, representing P(M∞, α, rnc, φ, CT ) = P(4, 0◦, 0.85, 0◦, 1) shows the areas of overpressure (yellowand orange) achieved on the capsule face which lead to a drag increase of 10.9%. To aid in visualizationof the pertinent flow physics leading to the drag augmentation values achieved in this study, a quad-nozzlestudy was completed and is described in the next section.

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aug

Freestream Mach Number, M!

Dra

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aug

Freestream Mach Number, M!

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ug

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n, D

aug

Freestream Mach Number, M!

Dra

g A

ug

menta

tio

n, D

aug

Freestream Mach Number, M!

rnc = 0.20

= 0° = 60°

rnc = 0.85

Figure 19. Drag augmentation values for the augmented tri-nozzle study.

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C. Quad-nozzle Study

The previous tri-nozzle study identified several cases of significant drag augmentation. To further study thephysical mechanisms leading to overpressure and drag augmentation, a similar parametric study is completedhere using a quad-nozzle model. The additional symmetry provided by the quad-nozzle configuration aidsgreatly in flowfield visualization. Examples from this study guide a discussion in Section IIID of the physicalmechanisms responsible for the drag increase.

1. Geometry and ParameterizationTable 4. Valuesof tested parametricvalues for the quad-nozzle study.

M∞ 2, 4, 6α 0◦, −5◦

rnc 0.5, 0.85φ 0◦, 30◦

CT 0, 0.5, 1, 3

The baseline configuration used for this study utilizes similar geometriesh

to those described in the tri-nozzle studies of Section IIIB. However, wenow study a configuration with 4 equally-spaced nozzles. The parametricspace studied here spans the same five dimensions as the tri-nozzle studies,P = P(M∞, α, rnc, φ, CT ), with tested values given in Table 4. In total, 78cases were run in this study.

2. Results and Analysis

Table 5 gives CD for the six baseline cases with no SRP which serve as reference values in the drag augmen-tation calculations (see Equation (10)). Drag augmentation values for each case are shown in the plots ofFigure 20, ranging from -1.01 to 0.12.

The most striking observation is that none of the configurations with nozzles at rnc = 0.50 produceda total drag augmentation. Even with the decreased jet exit pressures, the plumes in this study remainlarge enough to produce extensive wakes which result in low pressure and thus low drag on the capsuleface. Although several of the rnc = 0.5 cases do exhibit areas of overpressure on the capsule face, theextensive wake regions (similar to those in Figure 16) due to the interior positioning of the plumes counterthe overpressures and ultimately result in a decrease in the total drag of the body. Notably, certain casesactually resulted in 100% drag loss due to plume JI, which conveys that SRP, when acting as an aerospike,can seriously decrease the drag seen by an entering body. The remainder of the discussion will focus on theperiphery cases (rnc = 0.85).

Examining the effect of variation in Mach number shows that drag augmentation is never achieved inthe lowest Mach cases (M∞ = 2) in this study. Extensive flow visualization revealed that the plumes don’tpenetrate past the capsule bow shock until M∞ > 2, owing to the large shock stand-off distance at low Machnumbers. This suggests that the penetration of the jets past the capsule bow shock may play a part in thephysics leading to surface overpressures and drag augmentation.

A detailed look at one case, P(M∞, α, rnc, φ, CT ) = P(6, 0◦, 0.85, 0◦, 1) exhibiting 11.3% drag augmen-tation (with CD = 1.63), sheds light on the physical mechanisms involved. Figure 21 shows the capsuleface overpressures, pitch-plane Mach contours, and a close-up of the flow near the nozzle. The capsule faceplot shows CP with the color map again chosen such that white represents CP = CPoNS

i, overpressures(CP > CPoNS) are yellow and red, and underpressures (Cp < CPoNS) are blue. Plume and shock visualiza-tions are shown using a color map chosen as before such that the sonic line is white, supersonic flows areyellow to red, and subsonic flows are blue.

In this figure, we see that as a jet penetrates into the freestream flow, a bow shock system forms aroundthe plume itself, creating a plume-shock. Flow passing through the oblique section of this plume-shock isturned towards the capsule face and decelerated by the oblique shock with a subsequent decrease in stagnation

Table 5. CT = 0 (jets-off) reference values: CD

over a range of M∞ and α.

M∞ = 2 M∞ = 4 M∞ = 6α = 0◦ 1.523 1.473 1.465α = −5◦ 1.501 1.459 1.446

pressure. This Po decrease, however, is much less extremethan the decrease experienced by flow passing through thecapsule bow shock (PoOS > PoNS). In this way, flow hav-ing passed through the oblique section of the plume-shockis able to reach much higher pressures as it decelerates to-wards the capsule face, consistent with the overpressuresseen in the same figure.

hIn order to ameliorate the underexpansion of the plumes seen previously in the high CT cases, Pe is decreased to moreappropriate levels by further expanding the jets through an increase in Ae/A∗ to 20.4.

iCPoNS |M∞=6 ≈ 1.8

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Dra

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rnc = 0.50

= 0° = 30°

rnc = 0.85

Figure 20. Drag augmentation values for the quad-nozzle study.

Figure 22 shows iso-surfaces in the shock and plumes, offering a more three-dimensional understandingof the flow. This figure makes more apparent the oblique shock surfaces, or skirts, wrapping around each jet.These shock skirts, oblique to the flow, allow flow deceleration and compression through oblique rather thannormal shocks. With lower shock-losses, this flow recovers higher pressure values when it ultimately impingesupon the capsule face, yielding high-pressure areas on the surface behind each shock skirt. Predictably,the very center of the capsule face lying behind a normal shock at M∞ is white. The highest values ofoverpressure (Cp,max = 2.97 on the capsule face), seen as the yellow X-shaped pattern on the capsule face,occur between adjacent oblique shock skirts, implying that oblique shock-shock interactions have a role inthe drag augmentation achieved in this case. These observations led to the development of a flow model fordrag augmentation, described in more detail in Section IIID. At 2.97, the maximum overpressures on thecapsule face are fully 60% higher than CPoNS for this case which is only 1.82.

5

4

3

2

1

0

Mach

3.6

2.7

1.8

0.9

0

CP

Figure 21. Visualization of capsule face CP showing overpressures (CP > CPoNS) in yellow and red, Machcontours (sonic line in white with M > 1 in yellow and red and M < 1 in blue), and a zoomed view showingflow approaching the capsule: P(M∞, α, rnc, φ, CT ) = P(6, 0◦, 0.85, 0◦, 1).

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40

30

20

10

0

Pressure

3.6

2.7

1.8

0.9

0

CP

Figure 22. Three-dimensional representation of the plume-shocks, the pressure on a diagonal cutting-planeintersecting two adjacent nozzles, and an axial view showing the surface pressure coefficient with a “X”overpressure pattern due to shock-shock interactions of the plume-shock skirts. Plumes and plume-shocks arevisualized using isosurfaces of stagnation temperature, To.

Further examination of the 11 cases experiencing drag augmentation in this study indicated three differentoverpressure patterns, shown in Figure 23. As described previously in Section IIIB.2, a minimal amountof drag augmentation (2.3% in this case) can be achieved through flattening of the bow shock across thecapsule face (Figure 23a). Figure 23b displays the same oblique shock mechanism described in detail above,here with a drag augmentation of 4.9%. Figure 23c shows a typical case at α = −5◦ which achieved 8.9%

a) b) c)

5

4

3

2

1

0

Mach

3.6

2.7

1.8

0.9

0

CP

Figure 23. Various flows leading to drag augmentation in the quad-nozzle study. P = P(M∞, α, rnc, φ, CT ).

from the large overpressure occurringaround a single nozzle due to the obliqueshock skirt around the lower jet.

Analysis of P-variable trends rein-forces the proposed Daug mechanism. Aspreviously noted, low M∞ cases don’t seepenetration of the plumes past the cap-sule bow shock, thus preventing the for-mation of the plume-shock skirt. As ex-pected, no drag augmentation is seen inthese cases. Similarly, nozzles at hightilt angles (high φ) must be coupled withstrong jets (high CT ) to allow jet pen-etration past the capsule bow shock totrigger the Daug mechanism. In addition,configurations with large nozzle tilt an-gles see less Daug at α = 0◦ because theflow passing through the oblique shocksis projected onto the jet itself instead ofthe capsule face. However, holding thecapsule at an angle of attack is shown toyield overpressures around a single nozzleeven at high values of φ.

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plume shocks

Po1

PoOS

plume skirt-shock plume skirt-shock

Po3

slip line

Oblique Shock-Shock Interactions Typical Bow Shock

Po1

vs.

bow shock

reflected sh

ock reflected shock

PoBS

Po3 > PoBS

shock-shock

interactions

plume

skirt-shock

(a)

(b) (c)

Figure 24. Flow model for drag augmentation. a) Diagram showing plume-shocks resulting from retroplumes(not pictured). Green area represents capsule bow shock. Black lines indicate shock-shock interactions, in-cluding oblique interactions between adjacent plume-shock skirts. b) A two-dimensional diagram representinga slice through adjacent plume-shock skirts showing the interaction between two plume skirt-shocks (indi-cated by red-colored shocks in (a)). The resulting stagnation pressure Po3 is contrasted with c) the stagnationpressure behind a bow shock PoBS (similar to PoNS).

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D. Flow Model for Drag Augmentation

The preliminary parameter studies described in Sections IIIB and IIIC indicate the potential for significantdrag augmentation through supersonic retropropulsion. We now examine the underlying physical mechanismsresponsible for the overpressures established on the capsule face, highlighting the capability of this approachto produce meaningful drag augmentation in SRP systems.

Supersonic plumes behave in a manner similar to “hard” geometry. As the plume penetrates the super-sonic freestream, a bow shock system forms, wrapping around the jet plume as seen in Figure 24a. The skirtof this plume-shock is oblique to the oncoming flow, and freestream flow passing through the oblique section iscompressed without experiencing the massive stagnation pressure losses of a normal shock (PoOS > PoNS).In this way, portions of the capsule face surface can be protected from stagnation pressure losses by theoblique shock skirts of the plume-shocks. Moreover, interaction regions between adjacent plume-shock skirtslead to oblique shock-shock interactions (Figure 24b), further decelerating the oncoming flow and raising thepressure incrementally through multiple oblique shock compressions.

It is also noteworthy that this curtailment of stagnation pressure losses scales favorably with M∞, imply-ing that the drag benefits will increase with Mach number. Figure 25 gives Po2/Po1 values for both oblique

100

80

60

40

20

0

1

0.8

0.6

0.4

0.2

0

Po

2/P

o1 R

ati

os

Freestream Mach Number

Po

OS/P

oN

S

Figure 25. Stagnation pressure losses for normal(PoNS/Po1) and oblique shocks (PoOS/Po1), for γ = 1.4and a weak oblique shock with a 20◦ turning angle.PoOS/PoNS also plotted as a function of M∞.

and normal shocks as a function of M∞, and moreimportantly shows the increase in the ratio (obliqueto normal) of post-shock stagnation pressures withincreasing Mach number. For example, at M∞ = 3and γ = 1.4, the stagnation pressure loss behinda normal shock is about 70% (PoNS/Po1 = 0.328)while that behind a weak oblique shock with a 20◦

turning angle is only 20% (PoOS/Po1 = 0.796).Therefore, at this low Mach number, flow behind anoblique shock stagnates at a pressure 2 times greaterthan that behind a bow shock. Moreover, increas-ing M∞ to 6, at which PoNS/Po1 = 0.030, allows apost-oblique shock stagnation pressure of almost 13times greater than PoNS . At Mach 10, the ratio in-creases to 34, clearly illustrating this favorable Machscaling. From a trajectory standpoint, more drag athigher Mach numbers translates into increased timefor vehicle deceleration.

IV. Conclusion

Current Mars missions are already challenging the mass limits of existing EDL technology. With heavierpayloads on the horizon, future missions must undertake the development of new deceleration systems.This work used large-scale numerical parametric studies to examine the aerodynamic performance of abroad spectrum of supersonic retropropulsion systems and to identify key physical mechanisms that may beexploited for deceleration.

Three parametric studies were performed on tri- and quad-nozzle configurations, encompassing over 181cases. These studies spanned a Mach range from 2 to 8 and considered a range of thrust coefficients withretrojet nozzles located both near the center of the capsule face and towards the periphery at tilt angles from0◦ to 60◦. Drag data was presented for thrust coefficients ranging from 0 to 6. Drag augmentation valueson the order of 20% were observed, resulting from local surface pressures 60% higher than that achievedbehind a normal shock. Conversely, the results in much of the previous SRP literature were confirmed bymany of the parametric cases exhibiting a staggering decrease in drag (up to 104%), demonstrating thatwhen applied incorrectly, retrojets have the capacity to act as aerospikes.

This work proposed a physical model for SRP drag augmentation based on shock manipulation, in whichthe retrojets serve as oblique shock generators. With the plume-shock skirts protecting the capsule frommassive normal shock stagnation pressure losses, the flow approaching the capsule face is compressed bymultiple oblique shocks, leading to significant overpressure on the capsule face.

This model of drag augmentation is fully consistent with data from the parametric studies. By increasingthe influence of the shock skirts and creating more oblique-oblique interaction events, the stagnation pressure

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losses can be kept to a minimum whilst still establishing significant overpressures on the surface. In addition,minimizing the plume wake through control of jet size or placement allows for even greater drag values.This approach of oblique shock over normal shock compression also offers the potential for elevated dragaugmentation with increasing Mach number, allowing earlier deceleration of the entry system. Moreover,wielding SRP at higher altitudes would alleviate the need for stronger (more expensive) jets.

Since the proposed mechanism for improving deceleration is grounded in altering the capsule drag ratherthan increasing the jet thrust, retrojet fuel mass becomes less of an issue. With a fundamental physical mech-anism for drag augmentation identified, follow-on studies are planned to exploit this alternative approachto SRP implementation. Future work will include an optimization study and associated entry trajectoryanalysis in order to fully quantify the potential payoff in terms of delivered mass and establish the feasibilityof supersonic retropropulsion as a high-mass deceleration technique for future Mars EDL challenges.

Acknowledgments

N. Bakhtian thanks the Stanford Graduate Fellowship Program and the NSF Graduate Research Fel-lowship Program for fellowship support. We would like to thank Marian Nemec, Thomas Pulliam, JuanAlonso, and Ilan Kroo for many useful discussions. We also gratefully acknowledge the support of the NASAAdvanced Supercomputing Division for access to the Columbia and Pleiades superclusters upon which thesestudies were performed.

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19Aftosmis, M. J., Berger, M. J., and Melton, J. E., “Robust and Efficient Cartesian Mesh Generation for Component-BasedGeometry,” AIAA Journal , Vol. 36, No. 6, 1998, pp. 952–960.

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20Aftosmis, M. J., Berger, M. J., and Adomavicius, G., “A Parallel Multilevel Method for Adaptively Refined CartesianGrids with Embedded Boundaries,” AIAA Paper 2000-0808, 38th AIAA Aerospace Sciences Meeting, Reno, NV, January 2000.

21Nemec, M. and Aftosmis, M. J., “Adjoint Error Estimation and Adaptive Refinement for Embedded-Boundary CartesianMeshes,” AIAA Paper 2007-4187, 18th AIAA Computational Fluid Dynamics Conference, Miami, FL, June 2007.

22Nemec, M. and Aftosmis, M. J., “Adjoint Sensitivity Computations for an Embedded-Boundary Cartesian Mesh Method,”Journal of Computational Physics, Vol. 227, 2008, pp. 2724–2742.

23Nemec, M., Aftosmis, M. J., and Wintzer, M., “Adjoint-Based Adaptive Mesh Refinement for Complex Geometries,”AIAA Paper 2008-0725, 46th AIAA Aerospace Sciences Meeting, Reno, NV, January 2008.

24Venditti, D. A. and Darmofal, D. L., “Grid Adaptation for Functional Outputs: Application to Two-Dimensional InviscidFlows,” Journal of Computational Physics, Vol. 176, 2002, pp. 40–69.

25Aftosmis, M. J., Berger, M. J., and Murman, S. M., “Applications of Space-Filling Curves to Cartesian Methods forCFD,” AIAA Paper 2004-1232, 42nd AIAA Aerospace Sciences Meeting, Reno, NV, January 2004.

26Berger, M. J., Aftosmis, M. J., and Murman, S. M., “Analysis of Slope Limiters on Irregular Grids,” AIAA Paper2005-0490, 43rd AIAA Aerospace Sciences Meeting, Reno, NV, January 2005.

27Berger, M. J., Aftosmis, M. J., Marshall, D. D., and Murman, S. M., “Performance of a New CFD Flow Solver Using aHybrid Programming Paradigm,” Journal of Parallel and Distributed Computing, Vol. 64, No. 4, 2005, pp. 414–423.

28Aftosmis, M. J. and Rogers, S. E., “Effects of Jet-Interaction on Pitch Control of a Launch Abort Vehicle,” AIAA Paper2008-1281, 46th AIAA Aerospace Sciences Meeting, Reno, NV, January 2008.

29Aftosmis, M. J. and Nemec, M., “Exploring Discretization Error in Simulation-Based Aerodynamic Databases,” Pro-ceedings of the 21st International Conference on Parallel Computational Fluid Dynamics, edited by R. Biswas, DEStechPublications, Inc., May 2009.

30Murman, S. M., Aftosmis, M. J., and Nemec, M., “Automated Parameter Studies Using a Cartesian Method,” NASTechnical Report NAS-04-015, NASA Ames Research Center, Moffett Field, CA, November 2004.

31Wintzer, M., Nemec, M., and Aftosmis, M. J., “Adjoint-Based Adaptive Mesh Refinement for Sonic Boom Prediction,”AIAA Paper 2008-6593, 26th AIAA Applied Aerodynamics Conference, Honolulu, HI, August 2008.

32Nemec, M. and Aftosmis, M. J., “Computations of Aerodynamic Performance Databases Using Output-Based Refine-ment,” Invited lecture, 47th AIAA Aerospace Sciences Meeting, Orlando, FL, January 2009.

33Murman, S. M., Aftosmis, M. J., and Berger, M. J., “Numerical Simulation of Rolling-Airframes Using a Multi-LevelCartesian Method,” AIAA Paper 2002-2798, 20th AIAA Applied Aerodynamics Conference, St. Louis, MO, June 2002.

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